On a Hele–Shaw Type Domain Evolution with Convected Surface Energy Density: The Third‐Order Problem

Abstract

Abstract. We investigate a moving boundary problem with a gradient flow structure which generalizes Hele-Shaw flow driven solely by surface tension to the case of nonconstant surface tension coefficient taken along with the liquid particles at boundary. The resulting evolution problem is first order in time, contains a third-order nonlinear pseudodifferential operator and is degenerate parabolic. Well-posedness of this problem in Sobolev scales is proved. The main tool is the construction of a variable symmetric bilinear form so that the third-order operator is semibounded with respect to it. Moreover, we show global existence and convergence to an equilibrium for solutions near trivial equilibria (balls with constant surface tension coefficient). Finally, numerical examples in 2D and 3D are given.Key words. free boundary motion, degenerate nonlocal parabolic evolution AMS subject classifications. 35R35, 76B07DOI. 10.1137/0506269951. Introduction. It is the aim of the present paper to consider the generalization of the well-investigated Hele-Shaw flow problem to the case of nonconstant surface tension coefficient (or surface energy density). While experiments on such situations have been reported in the literature (e.g., [3,12]), theoretical investigations of this seem to be lacking. A first step in this direction has been made in [10] where shorttime solvability was proved for a Hele-Shaw problem with nonconstant surface tension coefficient and so-called kinetic undercooling. Here we discuss the problem without this regularization, using again the simple assumption that the surface energy density is convectively transported along the moving boundary.This leads to the following moving boundary problem: For a given bounded domain Ω(0) ⊂ R m and a given nonnegative function γ 0 defined on ∂Ω(0) one looks for a family of C 2 -domains Ω(t) ⊆ R m , t > 0 and functions ϕ(·, t) ∈ C 2 Ω(t) ,

Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation

“…There are many possible ways to study the Hele-Shaw equation: to mention a few approaches we quote various PDE methods based on L 2 -energy estimates (see the works of Chen [15], Córdoba, Córdoba and Gancedo [21], Knüpfer and Masmoudi [32], Günther and Prokert [29], Cheng, Granero-Belinchón and Shkoller [16]), there are also methods based on functional analysis tools and maximal estimates (see Escher and Simonett [26], the results reviewed in the book by Prüss and Simonett [38] and Matioc [34,35]) or methods using harmonic analysis tools and contour integrals (see the numerous results reviewed in the survey papers by Gancedo [27] or Granero-Belinchón and Lazar [28]). For the related Muskat equation (a two-phase Hele-Shaw problem), maximum principles have played a key role to study the Cauchy problem, see [12,18,10,22] following the pioneering work of Constantin, Córdoba, Gancedo, Rodríguez-Piazza and Strain [17].…”

Convexity and the Hele–Shaw Equation

“…This property still holds in the case where h has limited regularity. Many results have been obtained since the pioneering works of Craig and Nicholls ( [16]; see also [29,20,25]). It is known that (see [1,26]), for any s > n/2+ 1,…”

“…There are many other possible approaches to study the Cauchy problem for the Hele-Shaw equation. One can study the existence of weak solutions, viscosity solutions or classical solutons; we refer the reader to [5,6,7,13,18,19,20,21,23,24,28]. These papers consider different formulations of the Hele-Shaw problem and we notice that, for rough solutions, it is not obvious to check that these formulations are equivalent.…”

Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation

“…There are many possible ways to study the Hele-Shaw equation: to mention a few approaches we quote various PDE methods based on L 2 -energy estimates (see the works of Chen [15], Córdoba, Córdoba and Gancedo [24], Knüpfer and Masmoudi [36], Günther and Prokert [33], Cheng, Granero-Belinchón and Shkoller [16]), there are also methods based on functional analysis tools and maximal estimates (see Escher and Simonett [30], the results reviewed in the book by Prüss and Simonett [42] and Matioc [38,39]) or methods using harmonic analysis tools and contour integrals (see the numerous results reviewed in the survey papers by Gancedo [31] or Granero-Belinchón and Lazar [32]). For the related Muskat equation (a two-phase Hele-Shaw problem), maximum principles have played a key role to study the Cauchy problem, see [12,18,10,26] following the pioneering work of Constantin, Córdoba, Gancedo, Rodríguez-Piazza and Strain [17].…”